Characteristic Scales in Galaxy Formation

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by  Avishai Dekel

Characteristic Scales in Galaxy Formation
Avishai Dekel1,2
1 The Hebrew Universityof Jerusalem, Israel
2 Institut d’Astrophysiqueand Observatoire deParis, France
4
0
Abstract. Recent data, e.g. from SDSS and 2dF, reveal a robust bi-modality in the
0
distribution of galaxy properties, with a characteristic transition scale at stellar mass
2
M∗ ∼3×1010M⊙ (nearL∗),correspondingtovirialvelocity V ∼100kms−1.Smaller
n galaxies tendtobebluedisksof youngpopulations.They definea“fundamentalline”
Ja of decreasing surface brightness, metallicity and velocity with decreasing M∗, which
extendstothesmallest dwarfgalaxies. Galaxiesabovethecriticalscalearedominated
3
by red spheroids of old populations, with roughly constant high surface brightensand
2
metallicity, andthey tendtohost AGNs.Aminimum inthevirial M/L is obtained at

thesamemagicscale.Thisbi-modalitycanbethecombinedimprintofseveraldifferent
1
physical processes. On smaller scales, disks are built by cold flows, and supernova
v
3 feedback is effective in regulating star formation. On larger scales, the infalling gas
0 is heated by a virial shock and star formation can be suppressed by AGN feedback.
5 Anotherfeedbackmechanism–gasevaporationduetophoto-ionization–mayexplain
1 theexistenceoftotallydarkhalosbelowV ∼30kms−1.Thestandardcoolingbarriers
0 areresponsibleforthelooseupperandlowerboundsforgalaxies:10<V <300kms−1.
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0
/ 1 Introduction
h
p
- Observations indicatefourcharacteristicscalesintheglobalpropertiesofgalax-
o
r ies, expressed either in terms of virial velocity V, or stellar mass M∗:
t
s • An upper limit at V ∼300kms−1, which is the lower bound for clusters.
a
: • A robust bi-modality scale at M∗ ≃3×1010M⊙ [15], corresponding to V ∼
v 100kms−1. This is comparable to L∗ of the Schechter luminosity function,
i
X markingtheexponentialupperboundfordiskgalaxies.Smallergalaxiestend
r to be blue, star-forming disks with the surface brightness and metallicity
a decreasingwith decreasingM∗ (LSB).Moremassivegalaxiesaredominated
byred,old-populationspheroidswithhighsurfacebrightnessandmetallicity
insensitive to M∗ (HSB) [15]. AGNs typically reside in halos above the bi-
modality scale [16]. The halo M/L shows a minimum at the same scale [35].
• Dark-dark halos (DDH) with no luminous components seem to dominate
the halo distribution below V ∼ 30kms−1. The relatively flat faint-end
luminosityfunctionandthesteepermassfunctionpredictedbythestandard
ΛCDM cosmology cannot be reconciled with the virial theorem and the
Tully-Fisherrelationobeyedbytheluminousgalaxies(§5).Also,mostdwarf
spheroidals in the Local Group lie in this range, while dwarf irregulars are
typically V >30kms−1 [22] [32] [13] [9] [34].
• A lower bound at V ∼10kms−1 for dwarf galaxies (see [9]).
2 Avishai Dekel
We summarize here how theory predicts the following characteristic scales:
• A loose upper bound for cooling (by bremsstrahlung) on a dynamical time
at M ∼1012−13M⊙, corresponding to V ∼300kms−1.
• Avirialshock-heatingscaleatM∗ ∼3×1010M⊙,orV ∼100kms−1.Inless
massive halos virial shocks cannot form and the gas flows cold into the disk
vicinity (§2).
• A supernova-feedback scale at V ∼ 100kms−1, below which the potential
wells are shallow enough for the energy fed to the ISM by supernova to
significantly heat the halo gas, or even blow it out (§3).
• A photo-ionization scale at V ∼ 30kms−1. Once the gas is continuously
ionized by the background UV flux, IGM gas cannot accrete into smaller
halos and gas already in such halos evaporates via a steady thermal wind
(§5).
• A lower bound for cooling (by atomic hydrogen) at V ∼10kms−1.
We briefly address the origin of these scales, and try to associate them with
the observed scales.
2 Shock-Heating Scale
The standardpicture of disk formationis that after the darkhalo virializes,the
infalling gasis shock heatednearthe virialradius.It thencools radiativelyand,
if the cooling time is shorter than the dynamical time, the gas accretes onto a
central disk and forms stars with the associated feedback. The upper-limit halo
mass for efficient cooling is Mcool ∼ 1012−13M⊙ [27] [33]. The common wisdom
used to be that this explains the upper bound for disk galaxies,but it has been
realized that the predicted scale is somewhat higher than the observed scale.
Fig. 1 shows the time evolution of the radii of Lagrangian gas shells in a
spherical system consisting of gas (with primordial composition) and dark mat-
ter,assimulatedbyBirnboim&Dekel[5]usinga1Dhydrocode.Notshownare
the dark-matter shells, which collapse, oscillate and tend to increase the gravi-
tational attraction exerted on the gas shells. The left panel focuses on massive
halos of ∼ 1012M⊙. Indeed, a shock exists near the virial radius at all times; it
gradually propagates outward, encompassing more mass in time. The hot post-
shock gas is in a quasi-static equilibrium, pressure supported against gravity.
The right panel focuses on halo masses smaller by an order of magnitude. An
interesting new phenomenon is seen here, where a stable shock forms and prop-
agatesfromthe disktowardthe virialradiusonlyaftera massof∼1011M⊙ has
virialized.Inless massivesystems,the coolingrateis faster thanthe infalltime,
making the “post-shock” gas unstable against gravitational infall and thus not
being able to support an extended shock.
This behavior can be understood via the following instability analysis of
the gas just behind the shock [5]. The standard equation of state is written as
P =(γ−1)ρewhereρandearethegasdensityandinternalenergy,andγ =5/3
for a mono-atomic gas. In the no-cooling case, the adiabatic index is defined as
Characteristic Scales in Galaxy Formation 3
Fig.1. Time evolution of the radii of Lagrangian gas shells (lines) in a spherical sim-
ulation of aprotogalaxy consisting of primordial gasand darkmatter.Temperatureis
markedbycolor.Ashockshowsupasasharpbreakintheflowlinesandasanabrupt
change in temperature. The lower discontinuity marks the “disk” radius, formed due
toan artificialcentrifugal force.(a)Amassivesystem,wherethevirialized massesare
intherange1011−1013M⊙.(a)Alessmassivesystem,1010−1012M⊙.Avirialshock
exists only in systems more massive than >1011M⊙.
γ ≡(∂lnP/∂lnρ) , and the system is known to be gravitationally stable once
ad
γ >4/3. When there is radiative cooling at a rate q, we define a new quantity:
dlnP ρq t
dyn
γ ≡ =γ− =γ− . (1)
eff
(cid:18)dlnρ(cid:19) ρ˙e t
cool
The second equality follows from energy conservation, e˙ = −PV˙ −q, plugged
intotheequationofstate.Notethatγ =γ whenq =0,andthatthedifference
eff
betweenthetwoisnothingbutthe ratioofcharacteristictimest /t ,where
dyn cool
t ≡ ρ/ρ˙ ≃ r /u, with r the shock radius and u the infall velocity into it.
dyn s s
Using the jump conditions acrossthe shock,one can expressγ in terms of the
eff
pre-shock gas quantities:
(γ+1)4 ρr Λ(T) µ 2γ−1
γ =γ− s , T = u2 , (2)
eff 6(γ−1)2 |u3| k N (γ+1)2
B A
where ρ is the pre-shock density.
In order to test for stability, we perturb a shell by r →r+δr and compute
the sign of the restoring force: δ¨r/δr. We assume a homologous infall δr = uδt
(supported by the finding in the simulations above), and balancing forces r¨ =
−ρ−1∇P −GM/r3 =0. This leads to
δ¨r/δr∝γ−4/3−2γ−1(γ−γ ) . (3)
eff eff
The system is stable when this expression is positive, namely for
2γ t γ+2/3
cool
γ >γ = , or > . (4)
eff crit γ+2/3 t γ2−4γ/3
dyn
4 Avishai Dekel
For γ = 5/3, we get γ = 10/7 replacing the γ = 4/3 of the adiabatic
crit crit
case. The critical scale is then characterized by t /t = 21/5 at the disk
cool dyn
vicinity, which translates to t /t ≃1 at the virial radius. Note that in our
cool dyn
notation t ≃ R/V at the virial radius, while the dynamical time used in the
dyn
order-of-magnitude arguments of [27] and [33] is several times larger.
When γ is computed using eq.(2) just abovethe disk radius ateverytime
eff
ofthesphericalsimulation,wefindthatthecriterionofeq.(4)indeedworksvery
well. Before the shock forms, it is well below γ and graduallyrising, reaching
crit
γ almostexactlywhentheshockstartspropagatingoutward.Itthenoscillates
crit
about γ with a decreasing amplitude, following the oscillations in the shock
crit
radius seen in Fig. 1.
When put in a cosmological setting, we find that the critical halo mass for
virial-shock heating is
Mshock ≃(1−6)×1011M⊙. (5)
The higher estimate is obtained for near solar metallicity. This scale is compa-
rable to the observed bi-modality scale, and is quite insensitive to redshift. To
seethis,weuse thespherical-collapsevirialrelationsatz:M ∝(1+z)−3/2V3 ∝
(1+z)3R3, and the general trend in the relevant range Λ ∝ T−1/2, to obtain
t /t ∝M independent of z. Since at z >2 most halos are of M <M ,
cool dyn shock
weconcludethatinmostformingdisksthe gasneverheatsto thevirialtemper-
ature – it flows cold all the way to the disk vicinity.
Although we are back to a condition involving t /t , which seems to re-
cool dyn
mindusoftheoriginalcriterionbyRees&Ostriker,itisnowbasedonanalyzing
theactualphysicalprocessthatcouldberesponsiblefortheheatingtothevirial
temperature,namelytheexistanceofavirialshock.Becausethedynamicaltime
involved in the current analysis is smaller than the one used in the original es-
timate by a factor of a few, our derived upper bound for halo masses in which
cold flows prevail is smaller by a similar factor.
Preliminary results from cosmological hydro simulations indicate that the
phenomenonisnotrestrictedtosphericalsystems,andthatourstabilitycriterion
may in fact be approximatelyvalid in the generalcase where the inflow is along
filaments.Fig.2 showssnapshotsfromEuleriansimulationsby A.Kravtsov[19]
[20], showing the gas temperature and flow fields in two protogalaxies: one of
M ≃ 3×1011M⊙ at z ≃ 4, and the other of M ≃2×1010M⊙ at z ≃ 9. While
the more massive galaxy shows a hot gas component at the virial temperature
behind a virial shock, the smaller galaxy has mostly cold gas inside the virial
radius. Also seen in the massive galaxy are cold streams penetrating the hot
mediumtowardthecentraldisk.SimilarresultshavebeenobtainedbyFradalet
al.[11],whoemphasisthefeedingofgalaxiesbycoldflowspreferentiallyatearly
epochs, corresponding to less massive galaxies. We are in a process of testing
our shock-stability criterion using cosmological simulations, and generalizing it
to the realistic case of elongated streams.
Beyond introducing a new scale in galaxy formation, the cold streaming to-
ward the disk, as opposed to the commonly assumed shock heating and slow
accretion, may have several interesting implications. For example:
Characteristic Scales in Galaxy Formation 5
Fig.2.Snapshotsfromcosmologicalhydrosimulations(courtesyofAndreyKravtsov),
showing the gas temperature and flow fields in two protogalaxies. (a) A halo of M ≃
3×1011M⊙atz≃4,showingavirialshockandheatedgaspenetratedbycoldstreams.
(b) A halo of M ≃2×1010M⊙ at z ≃9, showing cold infall with little shock heating
inside thevirial radius of ∼7h−1kpc.
• Itmaymodifythestarformationrateandtheassociatedsupernovafeedback.
If the result of the cold stream hitting the disk is a burst of stars at high z,
thismayhelpthesemi-analyticmodels(SAMs)explainbetterthepresenceof
luminousredgalaxiesatz ∼1.Thisprocessshouldbeproperlyincorporated
in the SAMs.
• ItmayreducethesoftX-rayfluxfromhalos.Thisisconsistentwithobserved
individualgalaxies[6],andmayexplainwhytheobservedX-raybackground
is lower than the predictions based on shock heating in all halos [26].
• The cold gas may emit a significant fraction of its infall energy in Ly-α,
eitherifthestreamscoolandmergeintotherotatingdiskgraduallywithout
shocking [17], or after shockingat the disk vicinity.In the latter, the X-rays
produced are likely to be capturedinside a dense Strongremsphere of a few
kpc and the energy may eventually be released by Ly-α. This may explain
the observed Ly-α emitters [21].
3 Supernova Feedback Scale
Supernova feedback, which is a very different physical process, gives rise to a
characteristic scale in the same ballpark. The energy fed to the ISM by super-
novae can be written as
ESN ≃νǫM˙∗trad ∝M∗(tcool/tdyn) , (6)
whereν isthenumberofsupernovaeperunitmassofformingstars,ǫisthetyp-
ical supernova energy, and t is the time available for the supernova remnant
cool
to share its energy with the medium during its adiabatic phase,before a signifi-
cant fractionof it is radiatedaway.The secondproportionality follows from the
crude assumption M˙∗ ∼M∗/tdyn. Dekel & Silk [8] pointed out that for halos in
6 Avishai Dekel
the relevant range, where the virial temperature is T ∼ 105K, the cooling rate
behaves roughly like Λ ∝ T−1, and then t /t ≃ 0.01 for all galaxies. This
cool dyn
means that, despite the significant radiative losses, the energy fed to the ISM
is ESN ∝ M∗. When this energy input is compared to the energy required for
significantly heating the gas or blowing it out, E ≃ M V2, one ends up
binding gas
with the critical scale for effective feedback,
V ≃100kms−1. (7)
SN
In shallower potential wells, the feedback from a burst of stars can significantly
suppress further star formation, and produce low-surface-brightness (LSB) and
dwarfgalaxies.We notethatVSN correspondstoM∗ ≃3×1010M⊙,comparable
to the observed bi-modality scale.
Tightcorrelationsbetweenthepropertiesofgalaxiesbelow3×1010M⊙ define
a “fundamental line” (FL) which stretches across five decades in luminosity
down to the smallest dwarfs. The mean scaling relations, involving M∗, V, the
metallicity Z and the surface brightness µ, are approximately [15] [9] [34] [31]
V ∝M0.2, Z ∝M0.4, µ∝M0.6 . (8)
∗ ∗ ∗
Note that the familiar Tully-Fisher relation at the bright end, V ∝ M0.3 [7],
∗
seems to flatten in the dwarf regime. Supernova feedback can explain the origin
ofthe FLinsimpleterms[9].Theaboveenergycriterion,ESN ∝M∗ ∝MgasV2,
withtheassumptionthattherelevantgasmassisaconstantfractionofthehalo
mass, M ∝M, immediately implies the relation driving the FL:
gas
M∗/M ∝V2 . (9)
Assuming spherical collapse to virial equilibrium, the halo virial quantities are
related via, M ∝ V3 ∝ R3. In the instantaneous recycling approximation, we
approximateZ ∝M∗/Mgas.Forthestellarradiusweadoptthestandardassump-
tion of angular-momentum conservation [10] [24], R∗ ∝ λR, with λ a constant
spin parameter, and use it in µ∝M∗/R∗2. Combining the relations from eq. (9)
andon,werecoverthescalingrelations,eq.(8).Thissuccessofthissimplestpos-
sible toy model indicates that supernova feedback could indeed be the primary
driver of the FL below the bi-modality scale, and it strengths the associationof
this process with the origin of the transition scale itself.
4 The Origin of Bi-Modality
InFig.3,weschematicallyputtogetherthephysicalscalesdiscussedsofarinthe
classic density-temperaturediagram.Galaxies areloosely confinedto the region
wheretcool <tff.Theline M∗,shock ∼3×1010M⊙ distinguishesbetweenmassive
galaxieswhichsuffer virialshockheating andlowermass galaxieswhere the gas
flows cold toward the disk. The line V ∼ 100kms−1 marks the transition from
deeppotentialwellswhereHSBsformtoshallowpotentialwellswheresupernova
feedback is effective and leads to LSBs and dwarfs along the FL.
Characteristic Scales in Galaxy Formation 7
Fig.3. the characteristic scales in the classic diagram of gas density versus virial
velocity.Thehorizontal(gray)curvemarks2-σhalosinΛCDM.TheV-shaped(green)
curve refers to t = t ; galaxies can form only above this line. (a) The shock-
cool dyn
heating scale. (b) The supernova feedback scale. The two scales are comparable, at
V ∼100kms−1 or M∗ ∼3×1010M⊙.
The two different physical processes happen to yield a similar characteristic
scale, and they thus combine to produce the bi-modality. A possible scenario is
as follows:
• Halos below M had cold infall to disks, possibly associated with bursts
shock
of star formation. Galaxies above this threshold are built from progenitors
that were below the threshold at high z and thus formed early disks, which
later merged to spheroids of red, old stars. Any halo gas in such massive
halos consists of a hot medium and possibly cold streams.
• InhalosbelowV ,theSNfeedbackregulatesthestar-formationrate,which
SN
leadstoblue,youngstarsandtheFL,M∗/M ∝V2,withM/Lrisingtoward
smallerhalos.TheSNfeedbackmayalsosuppressanyAGNactivityinthese
halos.
• Inmoremassivehalos,the gas,whichhasbeenheatedbyvirialshocks,may
be prevented from cooling by AGN feedback, causing M/L to rise toward
larger halos in proportion to the total-to-cold gas ratio. There seems to
be enough energy in AGNs, though the actual feedback mechanism is not
properly understood yet.
• Protogalaxies below M may be seen as Ly-α emitters rather then as
shock
X-raysources,whilethehotgasinmoremassivehalosstillproducesX-rays.
5 Photo-evaporation Scale
Some of the halos below V ∼30kms−1 must be totally dark, with no luminous
component.Marginalevidence is starting to emerge fromthe statistics of image
multiplicities in gravitational lenses [18], and independently from the following
comparisonbetweentheoryandthe observedfrequencyofdwarfgalaxies.While
8 Avishai Dekel
thefaint-endgalaxyluminosityfunctionisroughlyφ(L)∝L−1 (orsteeper),the
halo mass function predicted in ΛCDM is roughly ψ(M) ∝ M−2 (or flatter).
These cannot be reconciled with the virial relation M ∝ V3 and the Tully-
Fisher relation in the dwarf regime, L ∝ V5 (or flatter). If, however, only a
fraction f (V) of the small halos have a luminous component (with L/M ∝V2
L
asdiscussedabove),thentheobservationscanbereconciledoncef ∝V2.Thus,
L
if the dark-dark halos appear at V ∼ 30kms−1, we expect ∼ 90% of the halos
to be totally dark at the faint end, V ∼10kms−1.
SupernovaoutflowsalonearenotlikelytoproduceDDHs,bothbecausethey
must leave a trace of stars behind, and because they tend not to be equally
effectiveinalldirections[23].Rampressureduetooutflowsfromnearbygalaxies
may remove gas in some cases [28], but probably not enough for the purpose.
Ontheotherhand,the originofDDHs couldbeexplainedbyradiative feedback.
The first stars and AGNs produce UV flux which ionizes most of the gas in the
universe by z ∼ 10, and keeps it at T ∼ (1−2)×104K until z ∼ 2 or
ion end
so [2]. Further infall of gas is suppressed in halos below the Jeans scale, which
is crudely estimated to be about V ∼ 30kms−1 [12], but this does not explain
the complete removal of the gas already residing in those halos. A significant
fraction of T ≃ 104K gas is expected to escape instantaneously from halos of a
correspondingly shallower potential, V < 10kms−1 [1], but not from halos of
10<V <30kms−1.
Fig.4. Theevaporatedmassfractionasafunctionofvirialmassandvelocity(insert).
The curves on the left correspond to instantaneous expulsion, and the curves on the
right refer to steady evaporation, where the ionization is assumed to be efficient from
z = 8 till either z = 4 (solid) or z = 1 (dashed). The gas temperature is either 104K
(left)or2×104K(right).WhilehalfthegasislostinstantaneouslyfromV ∼10kms−1
halos,asteadywindcanremoveasimilarfractionofthegasfromV ∼30kms−1 halos.
Characteristic Scales in Galaxy Formation 9
However, if a continuous input of energy keeps the gas ionized for many
dynamical times, it can evaporate via a steady thermal wind even from deeper
potential wells. Once the gas is lost from the shallow edge of the potential well,
it is continuously replenished by interior gas, and so on. In Shaviv & Dekel [29]
weanalyzedthisprocessfollowingtheclassicsolar-windanalysis[25],andfound
that a significant fraction of the gas eventually evaporates from halos as large
as V ∼30kms−1. Fig. 4 summarizes the results of this analysis.
Thus, halos in the range 10<V <30kms−1 lose most of their gas after the
cosmologicalreionization.They may end up as gas-poor dwarf spheroidals with
an old stellar population [9] [29], or as DDHs (see also [30]).
Halos of V < 10kms−1 are not expected to form stars at all because of the
cooling barrier at T ∼ 104K (Fig. 3). Atomic line cooling is very efficient just
abovethis temperature,but the only cooling agentbelow 104K is molecular hy-
drogen,which cannotsurvivethe UV backgroundand is anywayvery inefficient
[14].
6 Conclusion
The following table summarizes the characteristic scales originating from the
physical processes discussed above and their possible association with the ob-
served scales. The classic argument of cooling on a vaguely-defined dynamical
time provides relatively loose upper and lower bounds for luminous galaxies.
The combination of (a) the newly discoveredscale distinguishing between virial
shockheatingandcoldflowsand(b)thefamiliarsupernovafeedbackscaleseems
to produce the robust bi-modality imprinted on the observedgalaxy properties,
and in particular the characteristic upper limit for disk galaxies near L∗. This
is provided that AGN feedback, or another process such as thermal conductiv-
ity, indeed prevents the shock-heated gas in massive halos from ever forming
stars.The details of the proposedscenarioare yet to be workedout (see also[3]
[4]). While supernova feedback can be the primary process responsible for the
“fundamentalline”ofLSBanddwarfgalaxies,theradiativefeedbackduetocos-
mologicalreionizationmayberesponsiblefortotallyevacuatingsmallhalosfrom
gas and producing dwarf spheroidals and dark-dark halos. Once these physical
10 Avishai Dekel
process are properly incorporated in semi-analytic models of galaxy formation,
they may help solving the apparent conflicts between theory and observation.
Acknowledgment: I thank my collaborators Yuval Birinboimi and Nir Shaviv,
andthoseresponsibleforthe 3Dhydrosimulations,R.Dave,N.Katz,D.Keres,
A. Kravtsov, & D. Weinberg. This research has been supported by ISF 213/02
and NASA ATP NAG5-8218.
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